Improved upper bounds for partial spreads

Abstract

A partial \((k-1)\)-spread in \({\text {PG}}(n-1,q)\) is a collection of \((k-1)\)-dimensional subspaces with trivial intersection. So far, the maximum size of a partial \((k-1)\)-spread in \({\text {PG}}(n-1,q)\) was known for the cases \(n\equiv 0\pmod k\), \(n\equiv 1\pmod k\), and \(n\equiv 2\pmod k\) with the additional requirements \(q=2\) and \(k=3\). We completely resolve the case \(n\equiv 2\pmod k\) for the binary case \(q=2\).

Mathematics Subject Classification

Notes

Acknowledgments

The author thanks the referees for carefully reading a preliminary version of this article and giving very useful comments on its presentation. The work of the author was supported by the ICT COST Action IC1104 and Grant KU 2430/3-1—Integer Linear Programming Models for Subspace Codes and Finite Geometry from the German Research Foundation.